Aircraft: LARGE TRANSPORT
Phase of flight: Initial Climb
Anomaly: ATC Issue All Types; Deviation / Discrepancy - Procedural Published Material / Policy

I was working the Coordinator position; a position between two departure sectors in the TRACON. Aircraft X departed on the RNAV departure. Aircraft X never checked in with the Departure position. The departure Controller called Tower and asked them to switch Aircraft X. After multiple tries with tower; the controller called Aircraft X on guard. At the same time Aircraft Y was an overflight at 9;500 feet in Aircraft X's flight path. I as the Coordinator called Tower on multiple occasions to contact Aircraft X and give him a climb for traffic. As a last ditch effort I looked over to the other Departure Controller's scope and asked if he was talking to Aircraft X. The trainee called Aircraft X and he responded. Unsure of what to do; the Departure trainer asked me what he should do. I responded with 'give him traffic; say something'. The controllers at that time had turned the Aircraft Y and fortunately did not have a loss.On multiple occasions aircraft depart on the wrong frequency. We have been told that the only way to fix the problem is to report it. We have a box sitting on top of our flight plan printer where we put strips of aircraft that came off on the wrong frequency. This problem continues to exist and nothing is being done. Recommendation is for Tower to issue departure frequencies to all aircraft on departure. There is obviously a deficiency in the Tower of assigning departures the correct frequency.After review of the recordings; Aircraft X checked in with the wrong Departure Controller out of 1;300 feet and was ignored by the controller. The trainee has had multiple issues while training. He has been briefly qualified and then quickly decertified on all sectors and has struggled to maintain separation during his recertification. He in my opinion is a safety risk and has brought Area controllers down with him. My personal opinion is that his training should be stopped; he is a safety risk and management is not listening to the qualified controllers around him that are afraid to work with him. I personally tried to get off the Coordinator position prior to this event knowing that he was getting a skill check from a mediocre qualified supervisor and felt uncomfortable working next to them.The Area Supervisor is barely proficient Departure Control qualified. He appeared uneasy on the sector alone prior to the skill check and was sitting at another scope writing notes during the time of the incident. He has apologized for not hearing Aircraft X check in and not totally paying attention. The supervisors in the TRACON are required to only maintain limited time on 2 positions in the area. This makes them unfamiliar and not proficient in the entire area. This lack of knowledge is a constant issue with supervisor's abilities to correctly monitor and assist operations on a day to day basis. My opinion and recommendation is that the Supervisor's should be qualified and proficient on all positions in the area at which they supervise. Showing a lack of knowledge and situational awareness has led to misjudged actions and distractions in the facility. This is a major safety issue at the TRACON. NCT Departure Control Coordinator reported a departure contacted the TRACON on the wrong frequency and neither the trainee or the Supervisor observing the trainee noticed the wrong aircraft had called them.